We will start with an overview of some of the prehistoric creatures that have been hypothesized to survive as cryptids. Various sightings of winged, superficially-reptilian flying creatures have prompted some to suggest that pterosaurs might still be alive, and are responsible for such reports. I find this hypothesis unlikely, not because I am too closed-minded to accept that large flying animals could have survived to the present without being discovered, but because the reports that we have simply do not match anything we know about pterosaur biology. Most reports are very vague, and describe generic bat or bird-like flying monsters. Pterosaurs were not flying monsters. They were highly-specialized, bizarre animals that probably did not resemble anything familiar to the average layperson. We now know that they were quadrupedal, probably had an endothermic physiology, and were covered in superficially fur-like integumentary structures known as pycnofibres. None of this matches up with the reports, and it is very suggestive that the reports seem to describe inaccurate depictions of pterosaurs -- i.e., the depictions that the average layperson seems to have in mind when they think of pterosaurs. Many of the reports also sound like either bats or birds.
Taking this into account, I hypothesize that "pterosaur" sightings most-likely consist of misidentified bats and birds. However, it is possible that some of the bats and birds could potentially be unknown species.
Another group of Mesozoic sauropsids said to have persisted discreetly into the 21st century is the sauropods. As anyone familiar with dinosaurs or cryptozoology will already know, an African cryptid known as the Mokele-Mbembe is thought by many to be a relict sauropod. I used to subscribe to this hypothesis in the past, but I have now realized that it is flawed. As with the pterosaurs, reports do not match up well with modern depictions of sauropod anatomy and behavior. Sauropods were terrestrial animals with legs held directly underneath their bodies. By contrast, Mokele-Mbembe is usually described as a semi-aquatic creature with legs on the sides of its body. This description is more reminiscent of a reptile such as a monitor lizard or a turtle, rather than a dinosaur. Therefore, if Mokele-Mbembe exists, I think it is unlikely to be a sauropod.
At this point, it probably seems like I am anti-PSP. But in reality, nothing could be further from the truth. I strongly believe that science is all about pragmatism, not dogmatism or ideology, which are the domain of politics. Therefore, I take a pragmatic approach to my cryptozoological research, and the PSP is no exception.
Case in point: Plesiosaurs.
One of the most well-known hypotheses within cryptozoology is that plesiosaurs have survived to the present-day, and are responsible for reports of unidentified long-necked animals in oceans and lakes around the world. I used to think this hypothesis was unlikely, for various reasons. However, now that I have done some more research, I have arrived at a different conclusion. I now feel that the plesiosaur hypothesis fits well with the cryptozoological data, and is actually quite a plausible hypothesis. For more information on this matter, see the article that I wrote about plesiosaurs in May: http://www.mysteriouszoology.blogspot.com/2014/05/plesiosaurs.html
In addition to plesiosaurs, several other marine prehistoric survivors have also been proposed in the cryptozoological literature, including mosasaurs and archaeocete whales. I currently do not see any good evidence for these animals' continued existence, although I remain open-minded about the whole situation.
To conclude this portion of the article, I currently find that, of all the proposed prehistoric survivors, plesiosaurs are the only ones that I think marshal a relatively compelling case. However, my opinion could very well change if further evidence comes to light in the future.
I will now discuss my stance on the general concept of the PSP. One of the greatest criticisms of the PSP is that it is unreasonable to suggest that prehistoric taxa could have survived into the modern era without leaving a conspicuous fossil record between the time of their presumed extinction and the present-day. While this argument certainly does have merit, I don't think it completely invalidates the PSP. There are numerous examples of ghost lineages; gaps in the fossil record. While the lack of an intervening fossil record is, indeed, a major stumbling block for the PSP, I don't think it can be completely ruled out.
So this is my opinion on the PSP. I believe that it can be useful, and should not be completely discarded. However, in most cases, it does not fit the available data.